Morse inequalities for the Koszul complex of multi-persistence

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.     

Multiple solutions for a Schrödinger–Bopp–Podolsky system with positive potentials

In this paper, we prove existence of solutions for a Schrödinger–Bopp–Podolsky system under positive potentials. We use the Ljusternick–Schnirelmann and Morse Theories to get multiple solutions with a priori given “interaction energy.”     

高阶Morse芽的存在性

本文研究了多元C∞函数芽环中高阶Morse芽的存在性问题.利用由函数芽的偏导数生成的理想和C∞函数芽上的右等价关系,获得了在C∞函数芽环中,除了二元C∞函数芽环中有三阶和四阶的Morse芽以后,不再存在其它的Morse芽.以致在三元以上的C∞函数芽环中Morse引理不能推广到较高阶的情形.     

一维Morse势动量算符矩阵元的通项公式

导出一维Morse势中动量算符矩阵元〈n|Ap^G2q|n′〉的通项计算公式.